Singular FBSDEs and Scalar Conservation Laws Driven by Diffusion Processes

TL;DR

This paper analyzes singular forward-backward stochastic differential equations with degenerate generators and non-smooth terminal conditions, revealing existence, uniqueness, and flow property failures at terminal time, with implications for mathematical models in emissions markets.

Contribution

It extends the theory of FBSDEs to cases with degenerate structures and non-smooth terminal conditions, showing existence, uniqueness, and flow property breakdowns.

Findings

Existence and uniqueness results hold for singular FBSDEs.

Flow property can fail at terminal time with a Dirac mass.

Markovian representation breaks down at the terminal time.

Abstract

Motivated by earlier work on the use of fully-coupled Forward-Backward Stochastic Differential Equations (henceforth FBSDEs) in the analysis of mathematical models for the CO2 emissions markets, the present study is concerned with the analysis of these equations when the generator of the forward equation has a conservative degenerate structure and the terminal condition of the backward equation is a non-smooth function of the terminal value of the forward component. We show that a general form of existence and uniqueness result still holds. When the function giving the terminal condition is binary, we also show that the flow property of the forward component of the solution can fail at the terminal time. In particular, we prove that a Dirac point mass appears in its distribution, exactly at the location of the jump of the binary function giving the terminal condition. We provide a detailed analysis of the breakdown of the Markovian representation of the solution at the terminal time.